Twin Peaks

For the second printing of Banchoff's book, Beyond the Third
Dimension: Geometry, Computer Graphics, and Higher Dimensions, in
1994, we had the opportunity to replace some of the computer-generated
artwork with versions produced by more modern software. This
sequence of slices of the graph of a function replaced ones that had
no shading; there the three-dimensional effect is much enhanced.
The slices show how the level set changes from two curves in the
slices in the upper images to a single curve in the lower right. The
change occurs at a "critical point" when the level set forms a
figure-8, shown here at the lower left.

These images were produced using software that I had written as a graduate
student at Brown University. This was an early predecessor of the
StageTools package
that I began developing while at the
Geometry Center and continue to improve here at Union.